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Theorem lnolin 21348
Description: Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnoval.1  |-  X  =  ( BaseSet `  U )
lnoval.2  |-  Y  =  ( BaseSet `  W )
lnoval.3  |-  G  =  ( +v `  U
)
lnoval.4  |-  H  =  ( +v `  W
)
lnoval.5  |-  R  =  ( .s OLD `  U
)
lnoval.6  |-  S  =  ( .s OLD `  W
)
lnoval.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lnolin  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( T `  (
( A R B ) G C ) )  =  ( ( A S ( T `
 B ) ) H ( T `  C ) ) )

Proof of Theorem lnolin
Dummy variables  u  t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnoval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 lnoval.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
3 lnoval.3 . . . . 5  |-  G  =  ( +v `  U
)
4 lnoval.4 . . . . 5  |-  H  =  ( +v `  W
)
5 lnoval.5 . . . . 5  |-  R  =  ( .s OLD `  U
)
6 lnoval.6 . . . . 5  |-  S  =  ( .s OLD `  W
)
7 lnoval.7 . . . . 5  |-  L  =  ( U  LnOp  W
)
81, 2, 3, 4, 5, 6, 7islno 21347 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  L  <->  ( T : X --> Y  /\  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) ) ) ) )
98biimp3a 1281 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T : X --> Y  /\  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( (
u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `  t
) ) ) )
109simprd 449 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) ) )
11 oveq1 5881 . . . . . 6  |-  ( u  =  A  ->  (
u R w )  =  ( A R w ) )
1211oveq1d 5889 . . . . 5  |-  ( u  =  A  ->  (
( u R w ) G t )  =  ( ( A R w ) G t ) )
1312fveq2d 5545 . . . 4  |-  ( u  =  A  ->  ( T `  ( (
u R w ) G t ) )  =  ( T `  ( ( A R w ) G t ) ) )
14 oveq1 5881 . . . . 5  |-  ( u  =  A  ->  (
u S ( T `
 w ) )  =  ( A S ( T `  w
) ) )
1514oveq1d 5889 . . . 4  |-  ( u  =  A  ->  (
( u S ( T `  w ) ) H ( T `
 t ) )  =  ( ( A S ( T `  w ) ) H ( T `  t
) ) )
1613, 15eqeq12d 2310 . . 3  |-  ( u  =  A  ->  (
( T `  (
( u R w ) G t ) )  =  ( ( u S ( T `
 w ) ) H ( T `  t ) )  <->  ( T `  ( ( A R w ) G t ) )  =  ( ( A S ( T `  w ) ) H ( T `
 t ) ) ) )
17 oveq2 5882 . . . . . 6  |-  ( w  =  B  ->  ( A R w )  =  ( A R B ) )
1817oveq1d 5889 . . . . 5  |-  ( w  =  B  ->  (
( A R w ) G t )  =  ( ( A R B ) G t ) )
1918fveq2d 5545 . . . 4  |-  ( w  =  B  ->  ( T `  ( ( A R w ) G t ) )  =  ( T `  (
( A R B ) G t ) ) )
20 fveq2 5541 . . . . . 6  |-  ( w  =  B  ->  ( T `  w )  =  ( T `  B ) )
2120oveq2d 5890 . . . . 5  |-  ( w  =  B  ->  ( A S ( T `  w ) )  =  ( A S ( T `  B ) ) )
2221oveq1d 5889 . . . 4  |-  ( w  =  B  ->  (
( A S ( T `  w ) ) H ( T `
 t ) )  =  ( ( A S ( T `  B ) ) H ( T `  t
) ) )
2319, 22eqeq12d 2310 . . 3  |-  ( w  =  B  ->  (
( T `  (
( A R w ) G t ) )  =  ( ( A S ( T `
 w ) ) H ( T `  t ) )  <->  ( T `  ( ( A R B ) G t ) )  =  ( ( A S ( T `  B ) ) H ( T `
 t ) ) ) )
24 oveq2 5882 . . . . 5  |-  ( t  =  C  ->  (
( A R B ) G t )  =  ( ( A R B ) G C ) )
2524fveq2d 5545 . . . 4  |-  ( t  =  C  ->  ( T `  ( ( A R B ) G t ) )  =  ( T `  (
( A R B ) G C ) ) )
26 fveq2 5541 . . . . 5  |-  ( t  =  C  ->  ( T `  t )  =  ( T `  C ) )
2726oveq2d 5890 . . . 4  |-  ( t  =  C  ->  (
( A S ( T `  B ) ) H ( T `
 t ) )  =  ( ( A S ( T `  B ) ) H ( T `  C
) ) )
2825, 27eqeq12d 2310 . . 3  |-  ( t  =  C  ->  (
( T `  (
( A R B ) G t ) )  =  ( ( A S ( T `
 B ) ) H ( T `  t ) )  <->  ( T `  ( ( A R B ) G C ) )  =  ( ( A S ( T `  B ) ) H ( T `
 C ) ) ) )
2916, 23, 28rspc3v 2906 . 2  |-  ( ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )  ->  ( A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) )  ->  ( T `  ( ( A R B ) G C ) )  =  ( ( A S ( T `  B ) ) H ( T `
 C ) ) ) )
3010, 29mpan9 455 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( T `  (
( A R B ) G C ) )  =  ( ( A S ( T `
 B ) ) H ( T `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159    LnOp clno 21334
This theorem is referenced by:  lno0  21350  lnocoi  21351  lnoadd  21352  lnosub  21353  lnomul  21354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-lno 21338
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